Discussion on Floquet Theory: Mathieu, Hill and Related Equations

Why are we meeting?

Mathieu Equation:

z'' + [a - 2q cos(2ξ)] z = 0,

where prime(') indicates differentiation with respect to ξ.

Demonstrations for the Mathieu Equation:

Equation Governing the Magnetic Trap AFM of GRJ and PP:

z'' + 2ηz' + [a - 2q cos(2ξ)] z / [1 - κ cos(2ξ)] = 0,

where prime(') indicates differentiation with respect to ξ.

New parameters:

Various stability charts for this equation:

Eigenvalue locus for this equation: Animated Eigenvalue Locus

Conclusion

In order to understand what we see, we need to study Floquet's theory first.

Floquet's Theory: Study of Linear Homogeneous ODEs with Periodic Coefficients

Today, I will begin a discussion on this topic.


Example Matlab Code

Example Matlab Code

Objectives:

exampleCode
├── checkRK
│   ├── fnMSD.m
│   ├── fnMS.m
│   ├── msdRK1.m
│   ├── msdRK2.m
│   ├── msdRK4.m
│   ├── msRK1.m
│   ├── msRK2.m
│   ├── msRK4.m
│   ├── stepRK1.m
│   ├── stepRK2.m
│   └── stepRK4.m
└── mathieu
    ├── checkStability.m
    ├── findSlowOscPeriod.m
    ├── monodromyM.m
    ├── showMathieuSolutions.m
    ├── slopeMathieu.m
    ├── solveMathieu.m
    └── stepRK4.m

Checking the Runge-Kutta Methods

Try the following in the checkRK subfolder.

msRK1(0.1, 0.2, 1, 0, 10, 400)

msRK2(0.1, 0.2, 1, 0, 10, 200)

msRK4(0.1, 0.2, 1, 0, 10, 100)

Which of the three numerical solutions is the most accurate?

Applying Floquet Theory

Try the following in the mathieu subfolder.

showMathieuSolutions(0, 0.1, 100, 1000);

showMathieuSolutions(0, 0.5, 100, 1000);

showMathieuSolutions(0, 0.907, 200, 2000);

showMathieuSolutions(0.47, 0.5, 250, 2500);

showMathieuSolutions(0.471, 0.5, 250, 2500);

showMathieuSolutions(-0.121, 0.5, 250, 2500);

showMathieuSolutions(-0.122, 0.5, 250, 2500);


checkStability(0, 0.1)
checkStability(0, 0.5)
checkStability(0, 0.907)
checkStability(0, 0.91)
checkStability(0.47, 0.5)
checkStability(0.471, 0.5)
checkStability(-0.121, 0.5)
checkStability(-0.122, 0.5)


showMathieuSolutions(0, 0.1, 100, 1000);
% 88.76
findSlowOscPeriod(0, 0.1)
% 88.683

showMathieuSolutions(0, 0.1, 100, 1000);
% 16.6
findSlowOscPeriod(0, 0.1)
% 16.81

Dehmelt's Approximation

Even though we have discussed numerical techniques for calculating the solution of the Mathieu equation and other quantities associated with it, for small values of a and q, there is an approximation due to Dehmelt which provides simple estimates for both the slow and the fast components of the solution. Dehmelt’s approximation also provides a great deal of insight on the behaviour of physical systems described by the Mathieu equation. A crude derivation of the Dehmelt approximation is presented next.

Derivation of Dehmelt's approximation

Comparison of Dehmelt's approximation with the actual solution